#7929446

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5035 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 32 points:

(1,2)(3,7)(4,6)(5,8)(10,14)(15,16)(17,18)(19,24)(20,21)(22,28)(23,25)(26,27)(29,30)(31,32);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,16)(13,14)(17,19,23,29)(18,24,25,30)(20,22,27,31)(21,28,26,32);(2,6)(5,7)(18,25)(19,29)(20,27)(28,32);(1,2)(3,5)(4,6)(7,8)(9,11,13,12)(10,15,14,16)(17,20,23,27)(18,26,25,21)(19,22,29,31)(24,32,30,28);(1,3)(2,5)(4,8)(6,7)(9,12,13,11)(10,16,14,15)(17,21,23,26)(18,27,25,20)(19,28,29,32)(22,24,31,30);(17,22)(18,28)(19,27)(20,29)(21,30)(23,31)(24,26)(25,32)

Polycyclic presentation:

2^9|b^2=i,d^2=gh,e^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7836642);